The correct answer is $403$. For a linear equation in one variable to have infinitely many solutions, the coefficients of the variable must be equal on both sides of the equation and the constant terms must also be equal on both sides of the equation. The given equation can be rewritten as $\frac{4\left(3x+7\right)}{4}-\frac{s}{13}=r\left(x-8\right)$, or $3x+7-\frac{s}{13}=r\left(x-8\right)$. Applying the distributive property to the right-hand side of this equation yields $3x+7-\frac{s}{13}=rx-8r$. For this equation to have infinitely many solutions, the coefficients of $x$ must be equal, so it follows that $3=r$. Additionally, the constant terms must be equal, which means $7-\frac{s}{13}=-8r$. Substituting $3$ for $r$ in this equation yields $7-\frac{s}{13}=-8\left(3\right)$, or $7-\frac{s}{13}=-24$. Adding $\frac{s}{13}$ to both sides of this equation yields $7=-24+\frac{s}{13}$. Adding $24$ to both sides of this equation yields $31=\frac{s}{13}$. Multiplying both sides of this equation by $13$ yields $403=s$. Therefore, if the equation has infinitely many solutions, the value of $s$ is $403$.

Choice D is correct. A line in the xy-plane that passes through the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ has slope $m$, where $m=\frac{y_2-y_1}{x_2-x_1}$, and can be defined by an equation of the form $y-y_1=m\left(x-x_1\right)$. One of the lines shown in the graph passes through the points $\left(8,0\right)$ and $\left(3,4\right)$. Substituting $8$ for $x_1$, $0$ for $y_1$, $3$ for $x_2$, and $4$ for $y_2$ in the equation $m=\frac{y_2-y_1}{x_2-x_1}$ yields $m=\frac{4-0}{3-8}$, or $m=-\frac{4}{5}$. Substituting $-\frac{4}{5}$ for $m$, $8$ for $x_1$ and $0$ for $y_1$ in the equation $y-y_1=m\left(x-x_1\right)$ yields $y-0=-\frac{4}{5}\left(x-8\right)$, which is equivalent to $y=-\frac{4}{5}x+\frac{32}{5}$. Adding $\frac{4}{5}x$ to both sides of this equation yields $\frac{4}{5}x+y=\frac{32}{5}$. Multiplying both sides of this equation by $-10$ yields $-8x-10y=-64$. Therefore, an equation of this line is $-8x-10y=-64$. Similarly, the other line shown in the graph passes through the points $\left(8,0\right)$ and $\left(3,2\right)$. Substituting $8$ for $x_1$, $0$ for $y_1$, $3$ for $x_2$, and $2$ for $y_2$ in the equation $m=\frac{y_2-y_1}{x_2-x_1}$ yields $m=\frac{2-0}{3-8}$, or $m=-\frac{2}{5}$. Substituting $-\frac{2}{5}$ for $m$, $8$ for $x_1$, and $0$ for $y_1$ in the equation $y-y_1=m\left(x-x_1\right)$ yields $y-0=-\frac{2}{5}\left(x-8\right)$, which is equivalent to $y=-\frac{2}{5}x+\frac{16}{5}$. Adding $\frac{2}{5}x$ to both sides of this equation yields $\frac{2}{5}x+y=\frac{16}{5}$. Multiplying both sides of this equation by $10$ yields $4x+10y=32$. Therefore, an equation of this line is $4x+10y=32$. So, the system of linear equations represented by the lines shown is $4x+10y=32$ and $-8x-10y=-64$.

Choice A is incorrect and may result from conceptual or calculation errors. 

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice C is correct. It’s given that the function $f$ is defined by $f\left(x\right)=25x+30$. Substituting $2$ for $x$ in this equation yields $f\left(2\right)=25\left(2\right)+30$, which is equivalent to $f\left(2\right)=50+30$, or $f\left(2\right)=80$. Therefore, the value of $f\left(x\right)$ is $80$ when $x=2$.

Choice A is incorrect. This is the value of $25\left(2\right)$, not $25\left(2\right)+30$.

Choice B is incorrect. This is the value of $25+2+30$, not $25\left(2\right)+30$.

Choice D is incorrect. This is the value of $\left(25+30\right)\left(2\right)$, not $25\left(2\right)+30$.

The correct answer is $\frac{3}{17}$. It’s given that line $j$ is perpendicular to line $k$ in the xy-plane. This means that the slope of line $j$ is the negative reciprocal of the slope of line $k$. The equation of line $k$, $y=-\frac{17}{3}x+5$, is written in slope-intercept form $y=mx+b$, where $m$ is the slope of the line and $b$ is the y-coordinate of the y-intercept of the line. It follows that the slope of line $k$ is $-\frac{17}{3}$. The negative reciprocal of a number is $-1$ divided by the number. Therefore, the negative reciprocal of $-\frac{17}{3}$ is $\frac{-1}{-{\displaystyle \frac{17}{3}}}$, or $\frac{3}{17}$. Thus, the slope of line $j$ is $\frac{3}{17}$. Note that 3/17, .1764, .1765, and 0.176 are examples of ways to enter a correct answer.

The correct answer is $3$. Adding the second equation to the first equation in the given system of equations yields $\left(2\left(8x\right)-2\left(8x\right)\right)+\left(4\left(7y\right)+4\left(7y\right)\right)=12+12$, or $8\left(7y\right)=24$. Dividing both sides of this equation by $8$ yields $7y=3$. Substituting $3$ for $7y$ in the first equation, $2\left(8x\right)+4\left(7y\right)=12$, yields$2\left(8x\right)+4\left(3\right)=12$, or $2\left(8x\right)+12=12$. Subtracting $12$ from both sides of this equation yields $2\left(8x\right)=0$. Dividing both sides of this equation by $2$ yields $8x=0$. Substituting $0$ for $8x$ and $3$ for $7y$ in the expression $8x+7y$ yields $0+3$, or $3$. Therefore, the value of $8x+7y$ is $3$.

Choice C is correct. Let a equal the number of 120-pound packages, and let b equal the number of 100-pound packages. It’s given that the total weight of the packages can be at most 1,100 pounds: the inequality represents this situation. It’s also given that the helicopter must carry at least 10 packages: the inequality represents this situation. Values of a and b that satisfy these two inequalities represent the allowable numbers of 120-pound packages and 100-pound packages the helicopter can transport. To maximize the number of 120-pound packages, a, in the helicopter, the number of 100-pound packages, b, in the helicopter needs to be minimized. Expressing b in terms of a in the second inequality yields , so the minimum value of b is equal to . Substituting for b in the first inequality results in . Using the distributive property to rewrite this inequality yields , or . Subtracting 1,000 from both sides of this inequality yields . Dividing both sides of this inequality by 20 results in . This means that the maximum number of 120-pound packages that the helicopter can carry per trip is 5.

Choices A, B, and D are incorrect and may result from incorrectly creating or solving the system of inequalities.

The correct answer is $17$. It’s given that $2x+3=9$. Multiplying each side of this equation by $3$ yields $3\left(2x+3\right)=3\left(9\right)$, or $6x+9=27$. Subtracting $10$ from each side of this equation yields $6x+9-10=27-10$, or $6x-1=17$. Therefore, the value of $6x-1$ is $17$.

Choice B is correct. Substituting $72$ for $f\left(x\right)$ in the given function yields $72=8x$. Dividing each side of this equation by $8$ yields $9=x$. Therefore, $f\left(x\right)=72$ when the value of $x$ is $9$.

Choice A is incorrect. This is the value of $x$ for which $f\left(x\right)=64$, not $f\left(x\right)=72$.

Choice C is incorrect. This is the value of $x$ for which $f\left(x\right)=512$, not $f\left(x\right)=72$.

Choice D is incorrect. This is the value of $x$ for which $f\left(x\right)=640$, not $f\left(x\right)=72$.

Choice A is correct. For the linear function $f$, it’s given that $f\left(7\right)=28$. Substituting $7$ for $x$ and $28$ for $f\left(x\right)$ in the given function yields $28=4\left(7\right)+b$, or $28=28+b$. Subtracting $28$ from each side of this equation yields $0=b$. Therefore, the value of $b$ is $0$.

Choice B is incorrect. Substituting $1$ for $b$ in the given function yields $f\left(x\right)=4x+1$. For this function, when the value of $x$ is $7$, the value of $f\left(x\right)$ is $29$, not $28$.

Choice C is incorrect. Substituting $4$ for $b$ in the given function yields $f\left(x\right)=4x+4$. For this function, when the value of $x$ is $7$, the value of $f\left(x\right)$ is $32$, not $28$.

Choice D is incorrect. Substituting $7$ for $b$ in the given function yields $f\left(x\right)=4x+7$. For this function, when the value of $x$ is $7$, the value of $f\left(x\right)$ is $35$, not $28$.

The correct answer is $33$. It’s given in the table that the coordinates of two points on a line in the xy-plane are $(k,13)$ and $(k+7,-15)$. The y-intercept is another point on the line. The slope computed using any pair of points from the line will be the same. The slope of a line, $m$, between any two points, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, on the line can be calculated using the slope formula, $m=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}$. It follows that the slope of the line with the given points from the table, $(k,13)$ and $(k+7,-15)$, is $m=\frac{-15-13}{k+7-k}$, which is equivalent to $m=\frac{-28}{7}$, or $m=-4$. It's given that the y-intercept of the line is $(k-5,b)$. Substituting $-4$ for $m$ and the coordinates of the points $(k-5,b)$ and $(k,13)$ into the slope formula yields $-4=\frac{13-b}{k-\left(k-5\right)}$, which is equivalent to $-4=\frac{13-b}{k-k+5}$, or $-4=\frac{13-b}{5}$. Multiplying both sides of this equation by $5$ yields $-20=13-b$. Subtracting $13$ from both sides of this equation yields $-33=-b$. Dividing both sides of this equation by $-1$ yields $b=33$. Therefore, the value of $b$ is $33$.

Choice D is correct. It’s given that at the beginning of the $1\text{st}$ week of the year there was $\$600$ in a savings account and Gabriella deposits $\$35$ in that savings account at the end of each week. Therefore, the amount of money, in dollars, in the savings account at the end of the $4\text{th}$ week of that year is $600+4\left(35\right)$, or $740$.

Choice A is incorrect. This is the amount of money, in dollars, that will be in the account at the end of the $4\text{th}$ week if Gabriella withdraws, rather than deposits, $\$35$ at the end of each week.

Choice B is incorrect. This is the amount of money, in dollars, that will be in the account at the end of the $1\text{st}$ week, not the $4\text{th}$ week.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice B is correct. A system of two linear equations in two variables, $x$ and $y$, has no solution when the lines in the xy-plane representing the equations are parallel and distinct. Two lines are parallel and distinct if their slopes are the same and their y-intercepts are different. The slope of the graph of the given equation, $3x=36y-45$, in the xy-plane can be found by rewriting the equation in the form $y=mx+b$, where $m$ is the slope of the graph and $\left(0,b\right)$ is the y-intercept. Adding $45$ to each side of the given equation yields $3x+45=36y$. Dividing each side of this equation by $36$ yields $\frac{1}{12}x+\frac{5}{4}=y$, or $y=\frac{1}{12}x+\frac{5}{4}$. It follows that the slope of the graph of the given equation is $\frac{1}{12}$ and the y-intercept is $\left(0,\frac{5}{4}\right)$. Therefore, the graph of the second equation in the system must also have a slope of $\frac{1}{12}$, but must not have a y-intercept of $\left(0,\frac{5}{4}\right)$. Multiplying each side of the equation given in choice B by $\frac{1}{4}$ yields $\frac{1}{12}x=y$, or $y=\frac{1}{12}x$. It follows that the graph representing the equation in choice B has a slope of $\frac{1}{12}$ and a y-intercept of $\left(0,0\right)$. Since the slopes of the graphs of the two equations are equal and the y-intercepts of the graphs of the two equations are different, the equation in choice B could be the second equation in the system.

Choice A is incorrect. This equation can be rewritten as $y=\frac{1}{4}x$. It follows that the graph of this equation has a slope of $\frac{1}{4}$, so the system consisting of this equation and the given equation has exactly one solution, rather than no solution.

Choice C is incorrect. This equation can be rewritten as $y=\frac{1}{12}x+\frac{5}{4}$. It follows that the graph of this equation has a slope of $\frac{1}{12}$and a y-intercept of $\left(0,\frac{5}{4}\right)$, so the system consisting of this equation and the given equation has infinitely many solutions, rather than no solution.

Choice D is incorrect. This equation can be rewritten as $y=\frac{1}{36}x+\frac{5}{4}$. It follows that the graph of this equation has a slope of $\frac{1}{36}$, so the system consisting of this equation and the given equation has exactly one solution, rather than no solution.

The correct answer is . One method for solving the system of equations for y is to add corresponding sides of the two equations. Adding the left-hand sides gives , or 4y. Adding the right-hand sides yields . It follows that . Finally, dividing both sides of by 4 yields or . Note that 3/2 and 1.5 are examples of ways to enter a correct answer.

Choice D is correct. Adding the corresponding sides of the two equations eliminates y and yields , as shown.

If (x, y) is a solution to the system, then (x, y) satisfies both equations in the system and any equation derived from them. Therefore, .

Choices A, B, and C are incorrect and may be the result of errors when solving the system.

Choice D is correct. It’s given that a veterinarian recommends that each day the rabbit should eat $25$ calories per pound of the rabbit’s weight, plus an additional $11$ calories. If the rabbit’s weight is $x$ pounds, then multiplying $25$ calories per pound by the rabbit’s weight, $x$ pounds, yields $25x$ calories. Adding the additional $11$ calories that the rabbit should eat each day yields $25x+11$ calories. It’s given that $c$ is the total number of calories the veterinarian recommends the rabbit should eat each day if the rabbit’s weight is $x$ pounds. Therefore, this situation can be represented by the equation $c=25x+11$.

Choice A is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat $25$ calories per pound of the rabbit’s weight.

Choice B is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat $25+11$, or $36$, calories per pound of the rabbit’s weight.

Choice C is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat $11$ calories per pound of the rabbit’s weight, plus an additional $25$ calories.

Choice C is correct. It's given that $f\left(x\right)$ is the total cost, in dollars, to lease a car from this dealership with a monthly payment of $x$ dollars. Therefore, the total cost, in dollars, to lease the car when the monthly payment is $\$400$ is represented by the value of $f\left(x\right)$ when $x=400$. Substituting $400$ for $x$ in the equation $f\left(x\right)=36x+\mathrm{1,000}$ yields $f\left(400\right)=36\left(400\right)+\mathrm{1,000}$, or $f\left(400\right)=\mathrm{15,400}$. Thus, when the monthly payment is $\$400$, the total cost to lease a car is $\$\mathrm{15,400}$. 

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. The value of k can be found using the slope-intercept form of a linear equation, , where m is the slope and b is the y-coordinate of the y-intercept. The equation can be rewritten in the form . One of the given points, , is the y-intercept. Thus, the y-coordinate of the y-intercept must be equal to . Multiplying both sides by k gives . Dividing both sides by gives .

Choices B, C, and D are incorrect and may result from errors made rewriting the given equation.

The correct answer is $\frac{1}{4}$. For an equation in slope-intercept form $y=mx+b$, $m$ represents the slope of the line in the xy-plane defined by this equation. It's given that line $\mathcal{l}$ is defined by $3y+12x=5$. Subtracting $12x$ from both sides of this equation yields $3y=-12x+5$. Dividing both sides of this equation by $3$ yields $y=-\frac{12}{3}x+\frac{5}{3}$, or $y=-4x+\frac{5}{3}$. Thus, the slope of line $\mathcal{l}$ in the xy-plane is $-4$. Since line $n$ is perpendicular to line $\mathcal{l}$ in the xy-plane, the slope of line $n$ is the negative reciprocal of the slope of line $\mathcal{l}$. The negative reciprocal of $-4$ is $-\frac{1}{\left(-4\right)}=\frac{1}{4}$. Note that 1/4 and .25 are examples of ways to enter a correct answer.

The correct answer is $6$. A system of two linear equations in two variables, $x$ and $y$, has no solution if the lines represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form, $Ax+By=C$ and $Dx+Ey=F$, are parallel if the coefficients for $x$ and $y$ in one equation are proportional to the corresponding coefficients in the other equation, meaning $\frac{D}{A}=\frac{E}{B}$; and the lines are distinct if the constants are not proportional, meaning $\frac{F}{C}$ is not equal to $\frac{D}{A}$ or $\frac{E}{B}$. The first equation in the given system is $\frac{3}{2}y-\frac{1}{4}x=\frac{2}{3}-\frac{3}{2}y$. Multiplying each side of this equation by $12$ yields $18y-3x=8-18y$. Adding $18y$ to each side of this equation yields $36y-3x=8$, or $-3x+36y=8$. The second equation in the given system is $\frac{1}{2}x+\frac{3}{2}=py+\frac{9}{2}$. Multiplying each side of this equation by $2$ yields $x+3=2py+9$. Subtracting $2py$ from each side of this equation yields $x+3-2py=9$. Subtracting $3$ from each side of this equation yields $x-2py=6$. Therefore, the two equations in the given system, written in standard form, are $-3x+36y=8$ and$x-2py=6$. As previously stated, if this system has no solution, the lines represented by the equations in the xy-plane are parallel and distinct, meaning the proportion $\frac{1}{-3}=\frac{-2p}{36}$, or $-\frac{1}{3}=-\frac{p}{18}$, is true and the proportion $\frac{6}{8}=\frac{1}{-3}$ is not true. The proportion $\frac{6}{8}=\frac{1}{-3}$ is not true. Multiplying each side of the true proportion, $-\frac{1}{3}=-\frac{p}{18}$,  by $-18$ yields $6=p$. Therefore, if the system has no solution, then the value of $p$ is $6$. 

Choice A is correct. It’s given that the window repair specialist charges $\$220$ for the first two hours of repair plus an hourly fee for each additional hour. Let $n$ represent the hourly fee for each additional hour after the first two hours. Since it’s given that $x$ is the number of hours of repair, it follows that the charge generated by the hourly fee after the first two hours can be represented by the expression $n\left(x-2\right)$. Therefore, the total cost, in dollars, for $x$ hours of repair is $f\left(x\right)=220+n\left(x-2\right)$. It’s given that the total cost for $5$ hours of repair is $\$400$. Substituting $5$ for $x$ and $400$ for $f\left(x\right)$ into the equation $f\left(x\right)=220+n\left(x-2\right)$ yields $400=220+n\left(5-2\right)$, or $400=220+3n$. Subtracting $220$ from both sides of this equation yields $180=3n$. Dividing both sides of this equation by $3$ yields $n=60$. Substituting $60$ for $n$ in the equation $f\left(x\right)=220+n\left(x-2\right)$ yields $f\left(x\right)=220+60\left(x-2\right)$, which is equivalent to $f\left(x\right)=220+60x-120$, or $f\left(x\right)=60x+100$. Therefore, the total cost, in dollars, for $x$ hours of repair is $f\left(x\right)=60x+100$.

Choice B is incorrect. This function represents the total cost, in dollars, for $x$ hours of repair where the specialist charges $\$340$, rather than $\$220$, for the first two hours of repair.

Choice C is incorrect. This function represents the total cost, in dollars, for $x$ hours of repair where the specialist charges $\$160$, rather than $\$220$, for the first two hours of repair, and an hourly fee of $\$80$, rather than $\$60$, after the first two hours.

Choice D is incorrect. This function represents the total cost, in dollars, for $x$ hours of repair where the specialist charges $\$380$, rather than $\$220$, for the first two hours of repair, and an hourly fee of $\$80$, rather than $\$60$, after the first two hours.

Choice D is correct. After using the auger for 5 hours, Hector had removed 24,000 – 19,350 = 4,650 bushels of corn from the storage bin. During the 5-hour period, the auger removed corn from the bin at a constant rate of bushels per hour. Assuming the auger continues to remove corn at this rate, after x hours it will have removed 930x bushels of corn. Because the bin contained 24,000 bushels of corn when Hector started using the auger, the equation 24,000 – 930x = 12,840 can be used to find the number of hours, x, Hector will have been using the auger when 12,840 bushels of corn remain in the bin. Subtracting 12,840 from both sides of this equation and adding 930x to both sides of the equation yields 11,160 = 930x. Dividing both sides of this equation by 930 yields x = 12. Therefore, Hector will have been using the auger for 12 hours when 12,840 bushels of corn remain in the storage bin.

Choice A is incorrect. Three hours after Hector began using the auger, 24,000 – 3(930) = 21,210 bushels of corn remained, not 12,840. Choice B is incorrect. Seven hours after Hector began using the auger, 24,000 – 7(930) = 17,490 bushels of corn will remain, not 12,840. Choice C is incorrect. Eight hours after Hector began using the auger, 24,000 – 8(930) = 16,560 bushels of corn will remain, not 12,840.

Choice A is correct. The x-intercept of a graph in the xy-plane is the point on the graph where $y=0$. It's given that function $h$ is defined by $h\left(x\right)=4x+28$. Therefore, the equation representing the graph of $y=h\left(x\right)$ is $y=4x+28$. Substituting $0$ for $y$ in the equation $y=4x+28$ yields $0=4x+28$. Subtracting $28$ from both sides of this equation yields $-28=4x$. Dividing both sides of this equation by $4$ yields $-7=x$. Therefore, the x-intercept of the graph of $y=h\left(x\right)$ in the xy-plane is $\left(-7,0\right)$. It's given that the x-intercept of the graph of $y=h\left(x\right)$ is $\left(a,0\right)$. Therefore, $a=-7$. The y-intercept of a graph in the xy-plane is the point on the graph where $x=0$. Substituting $0$ for $x$ in the equation $y=4x+28$ yields $y=4\left(0\right)+28$, or $y=28$. Therefore, the y-intercept of the graph of $y=h\left(x\right)$ in the xy-plane is $\left(0,28\right)$. It's given that the y-intercept of the graph of $y=h\left(x\right)$ is $\left(0,b\right)$. Therefore, $b=28$. If $a=-7$ and $b=28$, then the value of $a+b$ is $-7+28$, or $21$.

Choice B is incorrect. This is the value of $b$, not $a+b$.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of $-a+b$, not $a+b$.

Choice C is correct. Dividing all terms in the given equation by $4$ yields $\frac{4x}{4}-\frac{28}{4}=-\frac{24}{4}$, or $x-7=-6$. Therefore, the value of $x-7$ is $-6$.

Choice A is incorrect. This is the value of $4x-28$, not $x-7$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The equation of line $h$ can be written in slope-intercept form $y=mx+b$, where $m$ is the slope of the line and $\left(0,b\right)$ is the y-intercept of the line. It’s given that line $h$ contains the points $\left(18,130\right)$, $\left(23,160\right)$, and $\left(26,178\right)$. Therefore, its slope $m$ can be found as $\frac{160-130}{23-18}$, or $6$. Substituting $6$ for $m$ in the equation $y=mx+b$ yields $y=6x+b$. Substituting $130$ for $y$ and $18$ for $x$ in this equation yields $130=6\left(18\right)+b$, or $130=108+b$. Subtracting $108$ from both sides of this equation yields $22=b$. Substituting $22$ for $b$ in $y=6x+b$ yields $y=6x+22$. Since line $k$ is the result of translating line $h$ down $5$ units, an equation of line $k$ is $y=6x+22-5$, or $y=6x+17$. Substituting $0$ for $y$ in this equation yields $0=6x+17$. Solving this equation for $x$ yields $x=-\frac{17}{6}$. Therefore, the x-intercept of line $k$ is $\left(-\frac{17}{6},0\right)$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice A is correct. Let the economist’s model be the linear function , where Q is the demand, P is the selling price, m is the slope of the line, and b is the y-coordinate of the y-intercept of the line in the xy-plane, where . Two pairs of the selling price P and the demand Q are given. Using the coordinate pairs , two points that satisfy the function are and . The slope m of the function can be found using the formula . Substituting the given values into this formula yields , or  . Therefore, . The value of b can be found by substituting one of the points into the function. Substituting the values of P and Q from the point yields , or . Adding 10,000 to both sides of this equation yields . Therefore, the linear function the economist used as the model is . Substituting 55 for P yields . It follows that when the selling price is $55 per unit, the demand is 16,250 units.

Choices B, C, and D are incorrect and may result from calculation or conceptual errors.

The correct answer is $\frac{65}{4}$. The linear relationship between $x$ and $y$ can be represented by the equation $y=mx+b$, where $m$ and $b$ are constants. It's given in the table that when $x=-12$, $y=-45$. Substituting $-12$ for $x$ and $-45$ for $y$ in the equation $y=mx+b$ yields $-45=-12m+b$, which can be rewritten as $-45+12m=b$. It's also given in the table that when $x=6$, $y=45$. Substituting $6$ for $x$ and $45$ for $y$ in the equation $y=mx+b$ yields $45=6m+b$, which can be rewritten as $45-6m=b$. Substituting $-45+12m$ for $b$ in this equation yields $45-6m=-45+12m$. Adding $6m$ to both sides of this equation yields $45=-45+18m$. Adding $45$ to both sides of this equation yields $90=18m$. Dividing both sides of this equation by $18$ yields $5=m$, or $m=5$. Substituting $5$ for $m$, $-12$ for $x$, and $-45$ for $y$ in the equation $y=mx+b$ yields $-45=5(-12)+b$, or $-45=-60+b$. Adding $60$ to both sides of this equation yields $15=b$. Therefore, $m=5$ and $b=15$. Substituting $5$ for $m$ and $15$ for $b$ in the equation $y=mx+b$ yields $y=5x+15$. Thus, the equation $y=5x+15$ represents the linear relationship between $x$ and $y$. It’s also given that the graph of the linear equation representing this relationship passes through the point $\left(\frac{1}{4},a\right)$. Substituting $\frac{1}{4}$ for $x$ and $a$ for $y$ in the equation $y=5x+15$ yields $a=5\left(\frac{1}{4}\right)+15$, which is equivalent to $a=\frac{5}{4}+15$, or $a=\frac{65}{4}$. Note that 65/4 and 16.25 are examples of ways to enter a correct answer.

The correct answer is $5$. It's given that the equation $10x+15y=85$ represents the situation, where $x$ is the number of on-site training courses, $y$ is the number of online training courses, and $85$ is the total number of hours of training courses the apprentice has enrolled in. Therefore, $10x$ represents the number of hours the apprentice has enrolled in on-site training courses, and $15y$ represents the number of hours the apprentice has enrolled in online training courses. Since $x$ is the number of on-site training courses and $y$ is the number of online training courses the apprentice has enrolled in, $10$ is the number of hours each on-site course takes and $15$ is the number of hours each online course takes. Subtracting these numbers gives $15-10$, or $5$ more hours each online training course takes than each on-site training course.

Choice A is correct. Hiro purchased 4 shirts and each shirt cost x dollars, so he spent a total of 4x dollars on shirts. Likewise, Hiro purchased 2 pairs of pants, and each pair of pants cost y dollars, so he spent a total of 2y dollars on pants. Therefore, the total amount that Hiro spent was 4x + 2y. Since Hiro spent $86 in total, this can be modeled by the equation 4x + 2y = 86. Using the same reasoning, Sofia bought 3 shirts at x dollars each and 5 pairs of pants at y dollars each, so she spent a total of 3x + 5y dollars on shirts and pants. Since Sofia spent $166 in total, this can be modeled by the equation 3x + 5y = 166.

Choice B is incorrect and may be the result of switching the number of shirts Sofia purchased with the number of pairs of pants Hiro purchased. Choice C is incorrect and may be the result of switching the total price each person paid. Choice D is incorrect and may be the result of switching the total price each person paid as well as switching the number of shirts Sofia purchased with the number of pairs of pants Hiro purchased.

Choice B is correct. It’s given that the equation $d=16-\frac{x}{30}$ gives the estimated amount of diesel $d$, in gallons, that remains in the gas tank of the truck after being driven $x$ miles. Substituting $300$ for $x$ in the given equation yields $d=16-\frac{300}{30}$, which is equivalent to $d=16-10$, or $d=6$. Therefore, the estimated amount of diesel that remains in the gas tank of the truck when $x=300$ is $6$ gallons.

Choice A is incorrect. This is the estimated amount of diesel, in gallons, that will remain in the gas tank of the truck when $x=480$, not when $x=300$.

Choice C is incorrect. This is the estimated amount of diesel, in gallons, that will remain in the gas tank of the truck when $x=60$, not when $x=300$.

Choice D is incorrect. This is the estimated amount of diesel, in gallons, that will remain in the gas tank of the truck when $x=0$, not when $x=300$.

Choice D is correct. It’s given that the graphs of the given system of equations intersect at the point $\left(4,y\right)$. Therefore, $\left(4,y\right)$ is the solution to the given system. Multiplying the first equation in the given system by $-2$ yields $-2ax-2by=-144$. Adding this equation to the second equation in the system yields $\left(-2a+6\right)x+\left(-2b+2b\right)y=\left(-144+56\right)$, or $\left(-2a+6\right)x=-88$. Since $\left(4,y\right)$ is the solution to the system, the value of $a$ can be found by substituting $4$ for $x$ in this equation, which yields $\left(-2a+6\right)\left(4\right)=-88$. Dividing both sides of this equation by $4$ yields $-2a+6=-22$. Subtracting $6$ from both sides of this equation yields $-2a=-28$. Dividing both sides of this equation by $-2$ yields $a=14$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice A is correct. It's given that the store's sales of a certain Greek yogurt totaled $\mathrm{1,277.94}$ dollars last month. It's also given that the equation $5.48x+7.30y=\mathrm{1,277.94}$ represents this situation, where $x$ is the number of smaller containers sold and $y$ is the number of larger containers sold. Since $x$ represents the number of smaller containers of yogurt sold, the expression $5.48x$ represents the total sales, in dollars, from smaller containers of yogurt. This means that $x$ smaller containers of yogurt were sold at a price of $5.48$ dollars each. Therefore, according to the equation, $5.48$ represents the price, in dollars, of each smaller container.

Choice B is incorrect. This expression represents the total sales, in dollars, from selling $y$ larger containers of yogurt.

Choice C is incorrect. This value represents the price, in dollars, of each larger container of yogurt.

Choice D is incorrect. This expression represents the total sales, in dollars, from selling $x$ smaller containers of yogurt.

Choice A is correct. Dividing each side of the given equation by $7$ yields $\frac{7\left(2x-3\right)}{7}=\frac{63}{7}$, or $2x-3=9$. Therefore, the equation $2x-3=9$ is equivalent to the given equation and has the same solution.

Choice B is incorrect. This equation is equivalent to $7\left(2x-3\right)=392$, not $7\left(2x-3\right)=63$.

Choice C is incorrect. Distributing $7$ on the left-hand side of the given equation yields $14x-21=63$, not $2x-21=63$.

Choice D is incorrect. Distributing $7$ on the left-hand side of the given equation yields $14x-21=63$, not $2x-21=70$.

Choice A is correct. It's given that line $j$ is perpendicular to line $k$ in the xy-plane. It follows that the slope of line $j$ is the opposite reciprocal of the slope of line $k$. The equation for line $k$ is written in slope-intercept form $y=mx+b$, where $m$ is the slope of the line and $b$ is the y-coordinate of the y-intercept of the line. It follows that the slope of line $k$ is $3$. The opposite reciprocal of a number is $-1$ divided by the number. Thus, the opposite reciprocal of $3$ is $-\frac{1}{3}$. Therefore, the slope of line $j$ is $-\frac{1}{3}$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. The given point, $\left(8,2\right)$, is located in the first quadrant in the xy-plane. The system of inequalities in choice A represents all the points in the first quadrant in the xy-plane. Therefore, $\left(8,2\right)$ is a solution to the system of inequalities in choice A.

Alternate approach: Substituting $8$ for $x$ in the first inequality in choice A, $x>0$, yields $8>0$, which is true. Substituting $2$ for $y$ in the second inequality in choice A, $y>0$, yields $2>0$, which is true. Since the coordinates of the point $\left(8,2\right)$ make the inequalities $x>0$ and $y>0$ true, the point $\left(8,2\right)$ is a solution to the system of inequalities consisting of $x>0$ and $y>0$.

Choice B is incorrect. This system of inequalities represents all the points in the fourth quadrant, not the first quadrant, in the xy-plane.

Choice C is incorrect. This system of inequalities represents all the points in the second quadrant, not the first quadrant, in the xy-plane.

Choice D is incorrect. This system of inequalities represents all the points in the third quadrant, not the first quadrant, in the xy-plane.

Choice C is correct. Adding the second equation of the given system to the first equation yields $5x+\left(-4x+y\right)=15+\left(-2\right)$, which is equivalent to $x+y=13$. So the value of $x+y$ is $13$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of $-(x+y)$.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. It's given that the cost of renting a backhoe for up to $10$ days is $\$270$ for the first day and $\$135$ for each additional day. Therefore, the cost $y$, in dollars, for $x$ days, where $x\le 10$, is the sum of the cost for the first day, $\$270$, and the cost for the additional $x-1$ days, $\$135\left(x-1\right)$. It follows that $y=270+135\left(x-1\right)$, which is equivalent to $y=270+135x-135$, or $y=135x+135$.

Choice A is incorrect. This equation represents a situation where the cost of renting a backhoe is $\$135$ for the first day and $\$270$ for each additional day.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice A is correct. Subtracting $180$ from both sides of the given equation yields $5p=70$. Dividing both sides of this equation by $5$ yields $p=14$. Therefore, the value of $p$ that satisfies the equation $5p+180=250$ is $14$.

Choice B is incorrect. This value of $p$ satisfies the equation $5p+180=505$.

Choice C is incorrect. This value of $p$ satisfies the equation $5p+180=610$.

Choice D is incorrect. This value of $p$ satisfies the equation $5p+180=\mathrm{1,430}$.

Choice A is correct. It’s given that the front of the roller-coaster car starts rising when it’s 15 feet above the ground. This initial height of 15 feet can be represented by a constant term, 15, in an equation. Each second, the front of the roller-coaster car rises 8 feet, which can be represented by 8s. Thus, the equation gives the height, in feet, of the front of the roller-coaster car s seconds after it starts up the hill.

Choices B and C are incorrect and may result from conceptual errors in creating a linear equation. Choice D is incorrect and may result from switching the rate at which the roller-coaster car rises with its initial height.

The correct answer is 2.6. According to the model, the head width, in millimeters, of a worker bumblebee can be estimated by adding 0.6 to 4 times the body weight, in grams, of the bee. Let x represent the body weight, in grams, of a worker bumblebee and let y represent the head width, in millimeters. Translating the verbal description of the model into an equation yields . Substituting 0.5 grams for x in this equation yields , or . Therefore, a worker bumblebee with a body weight of 0.5 grams has an estimated head width of 2.6 millimeters. Note that 2.6 and 13/5 are examples of ways to enter a correct answer.

Choice C is correct. The solution to this system of linear equations is represented by the point that lies on both lines shown, or the point of intersection of the two lines. According to the graph, the point of intersection occurs when $x=2$ and $y=3$, or at the point $\left(2,3\right)$. Therefore, the solution $\left(x,y\right)$ to the system is $\left(2,3\right)$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. It's given that $3x-8=7$. Adding $8$ to both sides of this equation yields $3x=15$. Adding $8$ to both sides of this equation yields $3x+8=23$. Therefore, the value of $3x+8$ is $23$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of $x$, not $3x+8$.

Choice C is incorrect and may result from conceptual or calculation errors.

The correct answer is $\frac{1}{2}$. For an equation in slope-intercept form $y=mx+b$, $m$ represents the slope of the line in the xy-plane defined by this equation. It's given that line $p$ is defined by $4y+8x=6$. Subtracting $8x$ from both sides of this equation yields $4y=-8x+6$. Dividing both sides of this equation by $4$ yields $y=-\frac{8}{4}x+\frac{6}{4}$, or $y=-2x+\frac{3}{2}$. Thus, the slope of line $p$ is $-2$. If line $r$ is perpendicular to line $p$, then the slope of line $r$ is the negative reciprocal of the slope of line $p$. The negative reciprocal of $-2$ is $-\frac{1}{\left(-2\right)}=\frac{1}{2}$. Note that 1/2 and .5 are examples of ways to enter a correct answer.

Choice B is correct. Dividing each side of the given equation by $3$ yields $x-9=8$. Therefore, the value of $x-9$ is $8$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the value of $3x-27$, not $x-9$.

Choice D is incorrect and may result from conceptual or calculation errors.

Choice C is correct. It’s given that line $r$ is perpendicular to line $p$ in the xy-plane. This means that the slope of line $r$ is the negative reciprocal of the slope of line $p$. If the equation for line $p$ is rewritten in slope-intercept form $y=mx+b$, where $m$ and $b$ are constants, then $m$ is the slope of the line and $\left(0,b\right)$ is its y-intercept. Subtracting $18x$ from both sides of the equation $2y+18x=9$ yields $2y=-18x+9$. Dividing both sides of this equation by $2$ yields $y=-9x+\frac{9}{2}$. It follows that the slope of line $p$ is $-9$. The negative reciprocal of a number is $-1$ divided by the number. Therefore, the negative reciprocal of $-9$ is $\frac{-1}{-9}$, or $\frac{1}{9}$. Thus, the slope of line $r$ is $\frac{1}{9}$.

Choice A is incorrect. This is the slope of line $p$, not line $r$.

Choice B is incorrect. This is the reciprocal, not the negative reciprocal, of the slope of line $p$.

Choice D is incorrect. This is the negative, not the negative reciprocal, of the slope of line $p$.